Use the Intermediate Value Theorem 1.11 and Rolle's Theorem 1.7 to show thatthe graph of/(x) = x 3 + 2x + k crosses the x-axis exactly once, regardless of the value ofthe constant k

Summary of results from sections 4.1-4.3 VECTOR SPACE AND ITS DEFINITION A vector space consists of the following four components A set of vectors V A set of scalars F (either the set of all real numbers R or the set of all complex numbers C). A rule for adding vectors in V: A rule for multiplying vectors in V: Then V is a vector space over F with addition and multiplication if the following 10 axioms (A1 ▯ A10) hold. Axiom 1 Closure under addition: For each pair of vectors u and v in V , the sum u + v is also in V . Axiom 2 Closure under scalar multiplication: For each vector v in V , and each scalar k, the scalar multiple kv is also in V . Axiom 3 Existence of a zero vector in V : In V there is a vector, denoted by 0 and called the zero vector, satisfying